User:Benji520/sandbox

A Bandwidth Sharing game is a type of resource allocation game designed to model the real world allocation of bandwidth to many users in a network. This game is popular in game theory because the conclusions can be applied to real life networks. The game is described as follows:

The Game

 * $$n$$ players
 * each player $$i$$ has utility $$U_i(x)$$ for amount $$x$$ of bandwidth
 * user $$i$$ pays $$w_i$$ for amount $$x$$ of bandwidth and receives net utility of $$U_i(x)-w_i$$
 * the total amount of bandwidth available is $$B$$

We will also use assumptions regarding $$U_i(x)$$
 * $$U_i(x)\ge0$$
 * $$U_i(x)$$ is increasing and concave
 * $$U(x)$$ is continuous

The game arises from trying to find a price $$p$$ so that every player individually optimizes their own welfare. This implies every player must individually find $$argmax_xU_i(x)-px$$. Solving for the maximum yields $$U_i^'(x)=p$$.

The Problem
With this maximum condition, the game then becomes a matter of finding a price that satisfies an equilibrium. Such a price is called a market clearing price.

A Possible Solution
A popular idea to find the price is a method called Fair Sharing. In this game, every player $$i$$ is asked for amount they are willing to pay for the given resource denoted by $$w_i$$. The resource is then distributed in $$x_i$$ amounts by the formula $$x_i=(\frac{w_i}{\sum_jw_j})*(B)$$. This method yields an effective price $$p=\frac{\sum_jw_j}{B}$$. This price can proven to be market clearing thus the distribution $$x_1,...,x_n$$ is optimal. The proof is as so:

Proof
$$argmax_{x_i}U_i(x_i)-w_i$$

$$\implies argmax_{w_i}U_i(\frac{w_i}{\sum_jw_j}*B)-w_i$$

$$\implies U^'_i(\frac{w_i}{\sum_jw_j}*B)(\frac{1}{\sum_jw_j}*B-\frac{w_i}{(\sum_jw_j)^2}*B)-1=0$$

$$\implies U^'_i(x_i)(\frac{1}{p}-\frac{1}{p}*\frac{x_i}{B})-1=0$$

$$\implies U^'_i(x_i)(1-\frac{x_i}{B})=p $$

Comparing this result to the equilibrium condition above, we see that when $$\frac{x_i}{B}$$ is very small, the two conditions equal each other and thus, the Fair Sharing game is almost optimal.